I've looked up both and to be honest my inexpert eye is struggling to tell the difference between them! Is there a way of identifying one from the other? I realise Spiraea x billardii is a hybrid from Spiraea douglasii (is that the right way of saying it?), but I assume there must be some difference that I'm not picking up on?!

Looks like douglasii comes as one smooth flower head like one of those rocket shaped ice lollies whereas billardii comes as a collection of small round flower heads joined or nearly joined together. So I now think that I was nearly right and that Sprizhenry is totally right.